Solutions of incompressible hydrodynamic flow of liquid crystals

In this paper, we investigate the incompressible hydrodynamic flow of the nematic liquid crystals in dimension NN (N=2N=2 or 3). We obtain the local existence and uniqueness of the solution if the initial density ρ0≥0ρ0≥0. Particularly, if ρ0ρ0 has a positive bound from below, and N=2N=2, we get the global existence and uniqueness of the solution with small initial data.